In the lowest resolution @media query CSS I just set the '.panels-col-first' and '.panels-col-last' to { width: 99.9%; clear: both }.

In higher resolution layouts, when you want it to go back to 2-col layout just undo the override { width: 49.9%; clear: none }

Notes:

the % widths should add up to LESS than 100% if you want to support IE 6 and 7 as they have rounding errors. You can always do this in your special ie.css if you have one and really need that 0.2% back!

These classes should already be set to float left by the main Panels CSS, but you can override that too if needed.

I had the same issue. No need to code in css to resolve the problem.
--> Omega handle well panels in responsive design.

"Omega comes with a handful of panels layouts by default. If you enable
Panels on your site with the Omega theme also enabled, and you go to
add a new panel you will see Omega: 12, ( or 16 or 24) Column Grid
listed. Choose one of those, then choose a layout. I'm embarrassed
about how easy it was, and I never even knew!"